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## G = D4.5D8order 128 = 27

### 5th non-split extension by D4 of D8 acting via D8/C8=C2

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — D4.5D8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C8○D4 — D4○C16 — D4.5D8
 Lower central C1 — C2 — C4 — C2×C8 — D4.5D8
 Upper central C1 — C2 — C2×C4 — C8○D4 — D4.5D8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D4.5D8

Generators and relations for D4.5D8
G = < a,b,c,d | a4=b2=1, c8=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c7 >

Subgroups: 180 in 72 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×3], C8 [×2], C8 [×3], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8 [×2], C23, C16 [×2], C16, C2×C8, C2×C8, M4(2), M4(2) [×3], D8 [×3], SD16 [×2], Q16 [×3], C2×D4, C2×Q8, C4○D4, C4.D4, C4.10D4, C8.C4 [×2], C2×C16, C2×C16, M5(2), M5(2), SD32 [×2], C8○D4, C2×D8, C2×Q16, C8⋊C22, C8.C22, M5(2)⋊C2, C8.17D4, C8.4Q8, D4.4D4, D4.5D4, D4○C16, C2×SD32, D4.5D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C4○D8, C87D4, D4.5D8

Character table of D4.5D8

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 8G 16A 16B 16C 16D 16E 16F 16G 16H 16I 16J size 1 1 2 4 16 2 2 4 16 2 2 4 4 4 16 16 2 2 2 2 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ3 1 1 1 -1 -1 1 1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 -1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 1 -1 1 1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 linear of order 2 ρ7 1 1 1 -1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ8 1 1 1 1 -1 1 1 1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ9 2 2 -2 0 0 2 -2 0 0 2 2 -2 0 0 0 0 -2 -2 -2 -2 2 0 0 0 0 2 orthogonal lifted from D4 ρ10 2 2 2 2 0 2 2 2 0 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 0 2 -2 0 0 2 2 -2 0 0 0 0 2 2 2 2 -2 0 0 0 0 -2 orthogonal lifted from D4 ρ12 2 2 2 -2 0 2 2 -2 0 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 2 0 -2 2 -2 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 -√2 √2 -√2 √2 √2 orthogonal lifted from D8 ρ14 2 2 -2 -2 0 -2 2 2 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 -√2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ15 2 2 -2 2 0 -2 2 -2 0 0 0 0 0 0 0 0 √2 -√2 √2 -√2 √2 √2 -√2 √2 -√2 -√2 orthogonal lifted from D8 ρ16 2 2 -2 -2 0 -2 2 2 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 √2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ17 2 2 2 0 0 -2 -2 0 0 0 0 0 2i -2i 0 0 √2 -√2 √2 -√2 -√2 -√-2 √-2 √-2 -√-2 √2 complex lifted from C4○D8 ρ18 2 2 -2 0 0 2 -2 0 0 -2 -2 2 0 0 0 0 0 0 0 0 0 2i 2i -2i -2i 0 complex lifted from C4○D4 ρ19 2 2 2 0 0 -2 -2 0 0 0 0 0 -2i 2i 0 0 √2 -√2 √2 -√2 -√2 √-2 -√-2 -√-2 √-2 √2 complex lifted from C4○D8 ρ20 2 2 2 0 0 -2 -2 0 0 0 0 0 2i -2i 0 0 -√2 √2 -√2 √2 √2 √-2 -√-2 -√-2 √-2 -√2 complex lifted from C4○D8 ρ21 2 2 -2 0 0 2 -2 0 0 -2 -2 2 0 0 0 0 0 0 0 0 0 -2i -2i 2i 2i 0 complex lifted from C4○D4 ρ22 2 2 2 0 0 -2 -2 0 0 0 0 0 -2i 2i 0 0 -√2 √2 -√2 √2 √2 -√-2 √-2 √-2 -√-2 -√2 complex lifted from C4○D8 ρ23 4 -4 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 0 0 2ζ1613+2ζ1611 2ζ1615+2ζ169 2ζ165+2ζ163 2ζ167+2ζ16 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 0 0 2ζ165+2ζ163 2ζ167+2ζ16 2ζ1613+2ζ1611 2ζ1615+2ζ169 0 0 0 0 0 0 complex faithful ρ25 4 -4 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 2ζ1615+2ζ169 2ζ165+2ζ163 2ζ167+2ζ16 2ζ1613+2ζ1611 0 0 0 0 0 0 complex faithful ρ26 4 -4 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 2ζ167+2ζ16 2ζ1613+2ζ1611 2ζ1615+2ζ169 2ζ165+2ζ163 0 0 0 0 0 0 complex faithful

Smallest permutation representation of D4.5D8
On 32 points
Generators in S32
```(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)
(1 8 9 16)(2 15 10 7)(3 6 11 14)(4 13 12 5)(17 20 25 28)(18 27 26 19)(21 32 29 24)(22 23 30 31)```

`G:=sub<Sym(32)| (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,8,9,16)(2,15,10,7)(3,6,11,14)(4,13,12,5)(17,20,25,28)(18,27,26,19)(21,32,29,24)(22,23,30,31)>;`

`G:=Group( (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,8,9,16)(2,15,10,7)(3,6,11,14)(4,13,12,5)(17,20,25,28)(18,27,26,19)(21,32,29,24)(22,23,30,31) );`

`G=PermutationGroup([(1,13,9,5),(2,14,10,6),(3,15,11,7),(4,16,12,8),(17,21,25,29),(18,22,26,30),(19,23,27,31),(20,24,28,32)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)], [(1,8,9,16),(2,15,10,7),(3,6,11,14),(4,13,12,5),(17,20,25,28),(18,27,26,19),(21,32,29,24),(22,23,30,31)])`

Matrix representation of D4.5D8 in GL4(𝔽7) generated by

 0 6 5 1 3 0 5 6 3 3 6 1 1 6 3 1
,
 4 3 2 0 6 0 4 6 1 1 4 5 0 0 0 6
,
 2 0 4 5 5 2 3 5 5 2 1 4 4 4 5 3
,
 3 6 3 3 6 5 2 5 6 1 0 2 2 4 0 6
`G:=sub<GL(4,GF(7))| [0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[4,6,1,0,3,0,1,0,2,4,4,0,0,6,5,6],[2,5,5,4,0,2,2,4,4,3,1,5,5,5,4,3],[3,6,6,2,6,5,1,4,3,2,0,0,3,5,2,6] >;`

D4.5D8 in GAP, Magma, Sage, TeX

`D_4._5D_8`
`% in TeX`

`G:=Group("D4.5D8");`
`// GroupNames label`

`G:=SmallGroup(128,955);`
`// by ID`

`G=gap.SmallGroup(128,955);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,736,422,1123,360,2804,718,172,4037,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^8=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^7>;`
`// generators/relations`

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